Anomalous scaling and generalized Lyapunov exponents of the one-dimensional Anderson model

We introduce a family of localization lengths ${\xi }_q$ (related to generalized Lyapunov exponents in a transfer-matrix approach) for the one-dimensional discrete Schrödinger equation with diagonal disorder. We show that, at the band edge of the pure system and with a random bounded potential with zero average, there is a q-dependent crossover in the scaling of ${\xi }_q$ with the disorder amplitude ε: For ε≤ε¯(q)∼$q^{\mathrm{-}6}$, ${\xi }_q$∝${\epsilon }^{\mathrm{-}2/3}$; otherwise, ${\xi }_q$∝${\epsilon }^{\mathrm{-}1/2}$. The limit therefore reproduces the scaling with exponent -(2/3), whereas deviations from this scaling law appear at each fixed ε if q is sufficiently large. These results involve a ‘‘multifractal’’ structure of the asymptotic decay of the wave functions.